Complex Disordered Systems

Today

• Colloid-colloid interactions
  • van der Waals interactions
  • double layer interactions
  • Entropic interactions: the Asakura-Oosawa model of depletion

Interactions between colloids

We distinguished between fundamental and effective forces between colloids

• Effective interactions emerge from the fundamnetal ones and structure our description at the colloidal scale.
• Fluctuations of fundamental forces often at the origin of the effective forces.
• We focus on effective interactions (revise your electrostatics if needed)

van der Waals interactions

IUPAC definition:

The attractive or repulsive forces between molecular entities (or between groups within the same molecular entity) other than those due to bond formation or to the electrostatic interaction of ions or of ionic groups with one another or with neutral molecules. The term includes:

• dipole–dipole
• dipole-induced dipole
• London (instantaneous, spontaneous dipole-induced dipole) forces.

The term is sometimes used loosely for the totality of nonspecific attractive or repulsive intermolecular forces.

Reminder: Multipole expansion

The electrostatic potential of a charge distribution can be expanded as:

V(\mathbf{r})=\frac{1}{4 \pi \epsilon_0} \sum_{l=0}^{\infty} \sum_{m=-l}^l \frac{4 \pi}{2 l+1} \frac{Q_{l m}}{r^{l+1}} Y_l^m(\theta, \phi)

where Y_l^m(\theta, \phi) are spherical harmonics and Q_{l m}=\int r^{\prime l} Y_l^{m *}\left(\theta^{\prime}, \phi^{\prime}\right) \rho\left(\mathbf{r}^{\prime}\right) d^3 r^{\prime}:

• Q_{00} = Q (monopole - total charge)
• Q_{1m} terms give the dipole moment \mathbf{p}
• =Q_{2m} terms give the quadrupole moment
• Higher terms: octupole, hexadecapole, etc.

For large distances (r \gg molecular size), the potential is dominated by the lowest non-zero multipole moment.

van der Waals interactions: dipole-dipole

We are focusing on atoms that are globally neutral. The first non-trivial term is the the dipole.

Static dipolew are asymmetric distribution of charge stationary in time

The interaction energy between two dipoles \mu_1 and \mu_2 separated by a distance r is proportional to:

U(r) \propto-\frac{\mu_1 \mu_2}{r^3}

van der Waals interactions: dipole-induced dipole

A permanent dipole can induce a temporary dipole in a nearby neutral atom. The induced dipole then interacts with the permanent dipole.

The induced dipole itself is proportional to the electric field of the incoming dipole, and it is \mu_B \propto \mu_A E_A \propto \dfrac{\mu}{r^3}.

Then the interaction energy is just

U(r) \propto -\frac{\mu^2 \alpha}{r^6}

where \alpha is the polarizability of the neutral atom.

Induce dipole- induced dipole interactions

• Orbitals provide the average expected electron density at every point in space, \langle\rho(\mathbf{r})\rangle=|\psi(\mathbf{r})|^2

• Instantaneous fluctuations from the mean occur in time and depend on density-density correlations. The variance is non-uniform .

• This means that even when the density is neutral, on short timescales (femtoseconds, electronic transiiton times) the desnity distributions can be thought of as asymmetric.

These quanto-mechanical fluctuations again produce interactions known as London dispersion forces

The key insight is the 1/r^{6} decay of the interactionwhich decays much mor rapidly than Coulomb’s 1/r.

They are short-range in 3d, since the total energy scales like \sim \int V(r) r^{d-1} d r and 6>3

van der Waals attractive interaction between spheres

Take now two spherical colloids of radus R at distance H.

Integrating the London interaction over all volume elements yield the van der Waals attracive potential

whera A_H is the Hamaker constant and f(h / R)=\left[\frac{2 R^2}{h^2-4 R^2}+\frac{2 R^2}{h^2}+\ln \left(\frac{h^2-4 R^2}{h^2}\right)\right].

van der Waals attractive interaction between spheres

Double-layer interaction: structure

Colloids are often charged. In the solution there will be

• co-ions (same charge as the colloid) that will be pushed away from the colloid surface, while
• counter-ions (opposite charge) will accumulate at the surface.

Double-layer interaction: nonlinear Poisson-Bolztmann equation

When two colloids come close, the charges in their double layers will interact giving rise to a repulsive interaction. Not simply Coulombic interaction, as it is mediated by the other (opposite) charges in the medium.

This means in practice taking Poisson’s equation

\nabla^2 \psi(\mathbf{r})=-\frac{\rho_{\mathrm{tot}}(\mathbf{r})}{\varepsilon}= \frac{\rho_{\mathrm{fixed}}(\mathbf{r})+\rho_\mathrm{ions}(\mathbf{r})}{\varepsilon}

and making the ion concentration depend on the electrostatic potential \psi(\mathbf{r}) itself via a Boltzmann weight

\rho_{\mathrm{ions}}(\mathbf{r})= z e n_s \exp \left(-\frac{z e \psi}{k_B T}\right)

with n_s the far-away (bulk) concentration of ions of salt and ze the ion charge.

This yields the nonlinear Poisson-Bolztmann equation

\nabla^2 \psi(\mathbf{r})=-\frac{1}{\varepsilon}\left[\rho_{\text {fixed }}+ ze n_s \exp \left(-\frac{z e \psi}{k_B T}\right)\right] .

Double-layer interaction: screened repulsion

Linearisation around room temperature (e\psi\ll k_BT) yields the Debye–Hückel equation

\nabla^2 \psi= \psi/\lambda_D^2

where \lambda_D is yhe Debye length

\lambda_D=\sqrt{\frac{1}{8 \pi \lambda_B n_s}} itself dependent on the Bjerrumm length \lambda_B: the scale at which two elemementary charge have energy k_B T.

The final form of the potential is and exponential repulsion with scale \lambda_D

vdW + double layer = DLVO interaction

to sum up hat identical colloids in solution have two interactions of opposite sign

• a van der Waals components, typically attractive

• a double layer component repulsive in nature

The sum of the two gives rise to the DLVO (Derjaguin–Landau–Verwey–Overbeek) interaction which is an elementary model for colloid stability and aggregation.

🪁 Play with the potential

Excluded volume and generic atomistic interactions

• At very short distances the electonic clouds overlap.
• Electrons cannot share the same quantum state and must occupt different, higher energy states

→ Large energy costs in approaching atoms: steric repulsive interactions below a van der Waals radius.

A phenomenological model for a generic atomistic interaction potential is then

V_{\mathrm{LJ}}(r)=\frac{A_n}{r^n}-\frac{B_m}{r^m}

where the first term is the short range repulsion and the second a generalised attraction.

A convenient choice is the Lennard-Jones potential of scale \sigma (twice the vdW radius)

V_{\mathrm{LJ}}(r)=4 \varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6\right],

where m=6 is chosen to match London’s forces and n=12 is chosen to match experiments and computational convenience.

Atomistic, but at the heart of numerous coarse-grained models .

Exclusion and entropic forces

Steric repulsion (hindrance) means exlcuded volume: at thermal energy scales, an atom cannot be placed at the position of another atom.

Geometrical point of view:

• Excluded volume means that, neglecting attractions, atoms/molecules//macromolecules/colloids are spheres that cannot overlap.

• Distribute spheres in space:

  • volume exclusion means that some configurations are forbidden

Exclusion in one dimension

Exclusion means forbidden configurations.

Statistical mechanics approach: simplest system with the required ingredient, studied in detail.

• Hard identical particles on a line between two repulsive, impenetrable walls.

Macroscopic analogue:

Clothpins on a line

Exclusion in one dimension

Investigate the probability distribution \rho(x) of finding the centre of a particle at position x. Parameters:

• clothpin horizontal size d (only size that matters)
• length of the line L
• number of clothpins N

We can treat the problem algorthmically

Simulation of hard clothpins on a line

🪁 Play with the simulation

Simulation of hard clothpins on a line

• The key parameter is the packing fraction \phi = \dfrac{dN}{L} (non-dimensional)¹.

  • low \phi: almost uniform profile
  • intermediate \phi: small peaks start to show close to the walls
  • high \phi: strong peaks and modulation of the density along the line

Attraction via repulsion

What do the peaks mean? It means that it is more likely to find the particles close to the walls:

• the walls seem to attract the particles
• peaks and valleys indicating layering
• the interaction with the wall vanishes as we get away from it

The source of this effect is entropy:

• if the first and last particles stay close to the walls, the other particles have more available space in the middle
• this means that there are more configurations available for the rest of the particles if the extreme particles are getting more localised
• more configurations → larger entropy → lower free energy = depletion

F=U - TS=\cancel{U} - TS

No internal energy, only entropy, so temperature is only a scale parameter.

Asakura-Oosawa depletion potential

More realism: Mixture of colloids (yellow) and polymers (squiggly lines inside red circles). The depletion layers are as thick as the polymer radius (red circles) and are indicated with the dashes around the colloids. When the two layers do not overlap, the osmotic pressure due to the polymers on the colloids is balanced. When there is overlap, there is a region inaccessible to the polymers (purple) and the pressure is unbalanced, leading to aggregation.

Asakura-Oosawa depletion potential

• Depletion forces play a pivotal role in colloidal aggregation.

• The basic mechanisms are the same as in our 1D example.

• We are going to work thrpugh the details of a particualr model

  • Asakura-Oosawa depletion interaction

Asakura-Oosawa model

Sketch of derivation of AO potential

• In the absence of colloids, the entire volume V is accessible to the polymer V_{\rm accessible} = V

• Grand potential of the polymers \Omega = -k_BT e^{\mu_p/k_B T}V_{\rm accessible}

• Introduce one colloid: the polymers can not get closer than the distance R+\delta

Excluded volume around one polymer

• Excluded volume V_{\rm exclusion} = V_{\rm outer}-V_{\rm inner}= \dfrac{4\pi}{3}\left((R+\delta)^3-R^3\right)
• The accessible volume is now V_{\rm accessible} = V-V_{\rm exclusion}

Sketch of derivation of AO potential

• Introduce second colloid at distance r from first colloid.

• The accessible volume is V_{\rm accessible}^{\infty}=V-2V_{\rm exclusion}

• Two situations:

  1. \forall r >R_d = 2R+2\delta the accessible volume remains the same, constant \Omega(r)=\Omega_{\infty}
  2. For 2R<r<R_d the accessible volume changes: overlap between exclusion volumes .

A second colloid is introduced at a short distance r. The exclusion regions overlap (light blue), increasing the accesible volume.

Sketch of derivation of AO potential

• The overlap region can be calulated from standard geometrical properties of spherical caps

V_{\mathrm{overlap}}(r)=\dfrac{4 \pi}{3} R_d^3\left[1-\frac{3}{4} \frac{r}{R_d}+\frac{1}{16}\left(\frac{r}{R_d}\right)^3\right]

• The new accessible volume is

V_{\rm accessible}^\prime=V-(2V_{\rm exclusion}-V_{\mathrm{overlap}})

• Since \Omega = -k_BT e^{\mu_p/k_B T}V_{\rm accessible} the gran potential becomes more negative when the the colloids are close.

Sketch of derivation of AO potential

• Fre energy advantage from the overlapping excluded volumes

\begin{aligned} W_{\rm AO}(r) = \Omega(r)-\Omega^{\infty} & =-k_BT e^{\mu/k_B T}\left(V_{\rm accessible}(r)-V_{\rm accessible}(\infty)\right)\\ & = -k_BT e^{\mu/k_B T}\left[V-2V_{\rm exclusion}+V_{\mathrm{overlap}}(r)-(V-2V_{\rm exclusion})\right]\\ & = -k_BT e^{\mu/k_B T} V_{\mathrm{overlap}(r)}\\ & = -k_B T \rho_p V_{\mathrm{overlap}(r)} \end{aligned}

The result is an attractive depletion potential controlled by the geometry of the overlap

AO potential

Summary of main colloid-colloid interactions